Abstract
A modification of the parameterization method is proposed to solve a linear two-point boundary value problem for a Fredholm integro-differential equation. The domain of the problem is partitioned and additional parameters are set as the values of the solution at interior points of the partition subintervals. Definition of a regular pair consisting of a partition and chosen interior points is given. The original problem is transformed into a multipoint boundary value problem with parameters. For fixed values of parameters, we get a special Cauchy problem for a system of integro-differential equations on the subintervals. Using the solution to this problem, the boundary condition and continuity conditions of solutions at the interior mesh points of the partition, we construct a system of linear algebraic equations in parameters. It is established that the solvability of the problem under consideration is equivalent to that of the constructed system.
Similar content being viewed by others
REFERENCES
A. Bressan and W. Shen, ‘‘A semigroup approach to an integro-differential equation modeling slow erosion,’’ J. Differ. Equat. 257, 2360–2403 (2014). https://doi.org/10.1016/j.jde.2014.05.038
M. Dehghan, ‘‘Solution of a partial integro-differential equation arising from viscoelasticity,’’ Int. J. Comput. Math. 83, 123–129 (2006). https://doi.org/10.1080/00207160500069847
M. Dehghan and F. Shakeri, ‘‘Solution of parabolic integro-differential equations arising in heat conduction in materials with memory via He’s variational iteration technique,’’ Int. J. Numer. Methods Biomed. Eng. 26, 705–715 (2010). https://doi.org/10.1002/cnm.1166
V. Volpert, Elliptic Partial Differential Equations (Springer, Basel, 2014). https://doi.org/10.1007/978-3-0348-0813-2
M. I. Berenguer, D. Gamez, and A. J. Lopez Linares, ‘‘Fixed point techniques and Schauder bases to approximate the solution of the first order nonlinear mixed Fredholm–Volterra integro-differential equation,’’ J. Comput. Appl. Math. 252, 52–61 (2013). https://doi.org/10.1016/j.cam.2012.09.020
A. A. Boichuk and A. M. Samoilenko, Generalized Inverse Operators and Fredholm Boundary-Value Problems (VSP, Utrecht, Boston, 2004).
S. Kheybari, M. T. Darvishi, and A. M. Wazwaz, ‘‘A semi-analytical approach to solve integro-differential equations,’’ J. Comput. Appl. Math. 317, 17–30 (2017). https://doi.org/10.1016/j.cam.2016.11.011
V. Lakshmikantham and M. R. Rao, Theory of Integro-Differential Equations (Gordon Breach, London, 1995).
A. M. Wazwaz, Linear and Nonlinear Integral Equations: Methods and Applications (Higher Education Press, Beijing, Springer, Berlin, Heidelberg, 2011).
S. Yüzbasi, ‘‘A collocation method based on Bernstein polynomials to solve nonlinear Fredholm-Volterra integro-differential equations,’’ Appl. Math. Comput. 273, 142–154 (2016). https://doi.org/10.1016/j.amc.2015.09.091
T. K. Yuldashev, ‘‘On the solvability of a boundary value problem for the ordinary Fredholm integrodifferential equation with a degenerate kernel,’’ Comput. Math. Math. Phys. 59, 241–252 (2019). https://doi.org/10.1134/S0965542519020167
T. K. Yuldashev, ‘‘On inverse boundary value problem for a Fredholm integro-differential equation with degenerate kernel and spectral parameter,’’ Lobachevskii J. Math. 40 (2), 230–239 (2019). equations,’’ J. Comput. Appl. Math. 294, 342–357 (2016). https://doi.org/10.1134/S199508021902015X
T. K. Yuldashev, ‘‘Determination of the coefficient and boundary regime in boundary value problem for integro-differential equation with degenerate kernel,’’ Lobachevskii J. Math. 38 (3), 547–553 (2017). https://doi.org/10.1134/S199508021703026X
D. S. Dzhumabaev, ‘‘A method for solving the linear boundary value problem for an integro-differential equation,’’ Comput. Math. Math. Phys. 50, 1150–1161 (2010). https://doi.org/10.1134/S0965542510070043
D. S. Dzhumabaev, ‘‘Necessary and sufficient conditions for the solvability of linear boundary-value problems for the Fredholm integro-differential equations,’’ Ukr. Math. J. 66, 1200–1219 (2015). https://doi.org/10.1007/s11253-015-1003-6
D. S. Dzhumabaev, ‘‘On one approach to solve the linear boundary value problems for Fredholm integro-differential equations,’’ J. Comput. Appl. Math. 294, 342–357 (2016). https://doi.org/10.1016/j.cam.2015.08.023
D. S. Dzhumabaev, ‘‘New general solutions to linear Fredholm integro-differential equations and their applications on solving the boundary value problems,’’ J. Comput. Appl. Math. 327, 79–108 (2018). https://doi.org/10.1016/j.cam.2017.06.010
D. S. Dzhumabaev, ‘‘Criteria for the unique solvability of a linear boundary-value problem for an ordinary differential equation,’’ Comput. Math. Math. Phys. 29, 34–46 (1989). https://doi.org/10.1016/0041-5553(89)90038-4
Funding
This research is supported by the Ministry of Education and Science of the Republic of Kazakhstan, grant no. AP 05132486 and the Award ‘‘Best University Teacher 2019.’’
Author information
Authors and Affiliations
Corresponding authors
Additional information
(Submitted by E. K. Lipachev)
Rights and permissions
About this article
Cite this article
Dzhumabaev, D.S., Nazarova, K.Z. & Uteshova, R.E. A Modification of the Parameterization Method for a Linear Boundary Value Problem for a Fredholm Integro-Differential Equation. Lobachevskii J Math 41, 1791–1800 (2020). https://doi.org/10.1134/S1995080220090103
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S1995080220090103